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October 6, 2013

Unexpected Connections

Posted by Tom Leinster

On Wednesday I’ll give a half-hour talk to all the new maths PhD students in Scotland, called Unexpected connections. What should I put in it?

When British students arrive to do a PhD, they have already chosen a supervisor, and they have a fairly good idea of what their PhD topic will be. So, they’re in the mood to specialize. On the other hand, they’re obliged to take some courses, and here in Scotland the emphasis is on broadening — balancing that specialization by learning a wide range of subjects.

Some students don’t like this. They’re not undergraduates any more, they’ve decided what they want to work on, and they resent being made to study other things. My job is to enthuse them about the wider, wilder possibilities — to tell them about some of the amazing advances that have been made by bringing together parts of mathematics that might appear to be completely unconnected.

What are some compelling stories I could tell? What are your favourite examples of apparently disparate mathematical topics that have been brought together to extraordinary effect?

The more disconnected the topics seem to be, the better. Best of all would be stories that connect pure mathematics with either applied mathematics or statistics.

Posted at October 6, 2013 11:08 PM UTC

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Re: Unexpected Connections

Re: Unexpected Connections

The other areas that would be needed would be some computational algebra (Matlab based for instance) and for the theoretical side of things to push the subject forward, probability and statistics. The Stanford team is just one in the States. There is a book by Edelsbrunner and Harer that shows what is needed. It is called Computational Topology.
There are also links with the CHomp project (Computational Homology Project that is) with applications ro structures of materials, (there is a book published on this and a website, here.)

The general area does link into some of your own thoughts and techniques, Tom, as well as Simon’s. There is also a nice area called Shape Theory (not the sort that I did/do but that of David Kendall) This looks at configuration spaces of groups and monoids that arise as symmetries etc of data-points, and then applies that to classification problems in biological statistics. You end up looking at probability distributions on manifolds.(Again there is a book, or look at this.)

Another area to look at would be Formal Concept Analysis (as mentioned by Simon), although the amount of detail one can give is a bit sparse perhaps. Finally look over at the John’s Azimuth site for applications of Petri nets to biological systems, manufacturing, etc. comparing with their use in Linear Logic.

Re: Unexpected Connections

Re: Unexpected Connections

One of the currently very hot topics is compressed sensing, which began with ideas from probability and Banach space geometry and is resulting in faster MRIs, digital cameras which take pictures that can be refocused after being taken, and other such exciting things. (Though this field is moving so fast that now — less than ten years after its birth — it’s already making the transition from being “mathematics” to being “engineering”.) There are millions of links (well, maybe just hundreds of thousands) and it’s not easy to find much written for an audience with no background in signal processing, but here’s a link that might interest you.

My personal favorite unexpected mathematical connection is the (still unexplained, but nevertheless extremely fruitful) connection between eigenvalues of random matrices and zeros of the Riemann zeta function. This may be old news now — certainly everyone working in either random matrix theory or in number theory knows it well, even if they don’t work on this aspect of things — but nobody saw it coming ahead of time. Try googling “Dyson Montgomery Riemann” for many retellings of how that story got started.

Re: Unexpected Connections

Re: Unexpected Connections

We goofed up: should have invited Atiyah instead. Oh well.

The piece you link to is nice — thanks — but it doesn’t quite fit the bill for what I need to do. The problem we have at the moment is that pure students tend to choose just pure courses, and applied students choose just applied courses, etc., even if it means that they’re covering material they’ve already covered before. What we’d most like to do is persuade pure students of the benefit of learning some applied maths (or probability or statistics), and so on.

Re: Unexpected Connections

So that’s what you call specialization?

As a “pure” student, I can argue that in fact if I don’t want to specialize too much and become, say, one of those probabilists who don’t know and don’t want to learn anything about algebra and geometry, then I’m left with so much “pure” stuff to learn that it seems just impossible to pay attention to the applied stuff. Isn’t that a good reason for a “pure” student to choose “pure” subjects?

Re: Unexpected Connections

I think your argument is completely reasonable. The main point of my presentation was to try to open the minds of the new PhD students, many of whom may not realize how many connections there are between apparently far-flung parts of mathematics. I wanted them to at least seriously consider choosing courses they might think are disconnected from their main interests, even if they decide not to in the end.

The other factor is that typically these students have to take six modules, and our current total offering consists of six pure modules, six applied, and four in probability and statistics. (I know, labels such as “pure” and “applied” aren’t necessarily helpful, but that’s the way it looks.) So, for example, a student who regards themself as entirely pure might end up taking all six pure courses even if they have already covered the material in some of them. This is clearly a waste of time, so I wanted to encourage them to do otherwise.

Re: Unexpected Connections

Well, rather close to home there is something you might know about: magnitude of metric spaces. That connects category theory and combinatorics (the Euler characteristic of a poset) with metric spaces, biological diversity, entropy, differential geometry, integral geometry, analysis, and probably some other stuff.

Re: Unexpected Connections

Yes… I think that was ultimately the reason why I was fingered to do this. Probably I will say something about it in the talk, but there’s a limit to how often I can stand to hear myself say the same thing.

Re: Unexpected Connections

If you can count the theoretical side of computing science as maths… One thing that struck me as being remarkable, at least until I understood it (thanks) is how Kleisli categories are exactly what’s needed to handle computation and effects in functional programming. It seems “obvious” that simple equational reasoning can’t be used when you have mutable state (x used to be 3, and now it is 4), functions that can fail, input and output and so on. It’s a very pleasant surprise to find out that with the help of a little bit of category theory you can go right back to using everything you know from high school algebra to prove properties of programs.

Re: Unexpected Connections

Thanks. That kind of thing is definitely on my list of topics to talk about. I’ll be pushing the line that the term “applied mathematics” deserves to be interpreted much, much more broadly than it often is. The application of category theory to computer science is an excellent example, even though we don’t actually run any courses on the latter.

Re: Unexpected Connections

I’m also coming more from the computer science side of things, but we do have some doozies.

One of my favorites is what Russell O’Connor dubs the Gauss-Jordan-Floyd-Warshall-McNaughton-Yamada algorithm, though names like Roy, Warshall, Kleene, and Thompson could be included as well. That is, the Gauss–Jordan (1887) algorithm for solving linear algebra problems, the Roy–Floyd-Warshall (1959; 1962a–c) algorithm for computing all-pairs shortest path on weighted graphs (also for taking transitive closures of graphs/relations), and the McNaughton-Yamada (1960) algorithm for constructing finite automata from their outputs, are all in fact the exact same algorithm just with different choices of *-semiring (i.e. closed semiring, not semiring with involution). As far as reinventing the wheel, this one is up there with the constant rediscovery of calculus (with variants of Gaussian elimination going back to 150BCE–179CE as well as being rediscovered by Newton in 1670). And yet, unlike calculus, the connections here are often missed by modern audiences.

Similarly, it took a terrifyingly long time for for the A*-search algorithm (1968) popular in artificial intelligence/machine learning to finally cross the hall (A*-parsing, 2003) into natural language processing, despite the fact that the connection is glaringly obvious— the parsing problem in NLP requires the exact same kind of search as path-finding in ML. It may seem passe, since these days we recognize the huge overlap between NLP and ML, but evidently this was not always so obvious.

The connection between category theory and type theory is very fruitful these days— the connection between co/monads and effects being one of the more popular unexpected connections. The Curry–Howard correspondence is another big one; although it’s now widely known, it’s still a fruitful area for drawing connections between logic and programming. And the connection between domain theory and semantics is still alive; there’s the classic work by Dana Scott on handling partially-defined terms, but there’s also more recent work on things like deterministic parallelism (LVars/LVish).

Re: Unexpected Connections

How about this Real Life StoryTM{}^{\text{TM}}? When I was a PhD student at Edinburgh doing knots and topology, I went to an Edinburgh Mathematical Society talk in St Andrews on Solar Physics given by Mitchell Berger. He was trying to calculate braiding in magnetic field lines of solar flares by defining combinatorial ‘higher order winding numbers’. I realized that the rational homotopy theory and finite type invariants of braids that I’d be thinking about could be used to tackle this kind of problem. The resulting ideas ended up forming half of my thesis.

Re: Unexpected Connections

Here is a pair of separate applied problems that surprisingly have the same underlying combinatorial principles organizing their solutions. One is the problem of classifying colliding waves in a shallow puddle, the other of organizing the likely genetic histories of a collection of related species. Compare the pictures for a quick impression: Figure 1 of Levy and Pachter with Figure 34 of Kodama and Williams.

It was fun to hear the two papers presented in succession at a conference a couple of years ago: Pachter joked that he could simply recycle the graphics from the title slide from Williams’s talk.

Bonus: the colliding wave picture has something very much in common with Postnikov’s amplituhedron , but I am in no position to comment more precisely!

Re: Unexpected Connections

Thanks very much to everyone for your suggestions, including the ones I didn’t use (which was mostly out of ignorance — in order to stand up and talk about something in front of several dozen people, I need to have at least a tiny sense of knowing what I’m talking about).

In the end, I decided to make these my main topics:

algebraic topology of data sets (persistent homology etc.)

topology applied to DNA (un)knotting

random matrices and Riemann’s ζ\zeta

statistical inference as a branch of logic

homotopy type theory.

You can see my slides here, though I suspect they won’t entirely make sense without the extensive waffle that accompanied them.

Re: Unexpected Connections

Re: Unexpected Connections

A very fruiful source of unexpected connections is the subject of expander graphs. To someone who loves self-referentiality, it will be even more intruiging to know that expander graphs are themselves defined through their high connectivity!

My own understanding is way too shallow to explain anything in detail, so I can only quote from an expository paper of Lubotzky,

Various deep mathematical theories have been used to give explicit constructions, e.g., the Kazhdan property (T) from representation theory of semisimple Lie groups and their discrete subgroups, the Ramanujan Conjecture (proved by Deligne) from the theory of automorphic forms, and more. [..]

We now start to see applications in the opposite direction: from expander graphs to number theory. The most notable one is the development of the affine sieve method. [..]

We will describe several results about groups whose formulations
do not mention expanders but expanders come out substantially in the proofs. [..]

several ways in which expanders have appeared, somewhat unexpectedly, in geometry. Most of these applications are for hyperbolic manifolds.

The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded valence is precisely an asymptotic robust graph with number of edges growing linearly with size (number of vertices), for all subsets.

Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL=L and the PCP theorem. In cryptography, expander graphs are used to construct hash functions.

Re: Unexpected Connections

Very interesting thread, and nice slides.

Each of all these interesting connections looks
like worth a lifetime study. But there seem to
be thousands of such interesting connections!
Perhaps we should ask ourselves what are the
connections between all these unexpected
connections?

Cheers,
Joachim.

PS: I like Mike’s remark about infinity-groupoids.

PPS: Re 404: this means that the server did not
expect the connection.

Re: Unexpected Connections

I’m disappointed. I thought I was going to see a photo of an inebriated man acting like the life of the party; instead it looks like a man in native costume. (The slide has something on probability and statistics, and the ‘unexpected’ connection with the inebriated man wearing a lampshade could be the concept of a random walk.)

Re: Unexpected Connections

The day after I gave my talk, I got an email from science writer Monica Kortsha at the University of Texas, pointing me to a story she’d written on the interplay between numerical analysis and engineering — which contains a surprise.

Re: Unexpected Connections

As Eugene says, we’d need some more details to judge how surprising it is.

Perhaps also I’m just used to hearing many accounts of how we’re not as good as we once were at doing calculations, from the class room to Arnold’s criticism of what the modern mathematician can’t do quickly.

Re: Unexpected Connections

After a reflection, I think it should not be surprising. Not because engineers are sloppy or stupid but because numerical analysis is hard and deep. In fact we had a bit of a discussion about numerical analysis a few years ago.